Feller property and infinitesimal generator of the exploration process

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Feller property and infinitesimal generator of the exploration process

Romain Abraham, Jean-François Delmas

To cite this version:

Romain Abraham, Jean-François Delmas. Feller property and infinitesimal generator of the explo- ration process. Journal of Theoretical Probability, Springer, 2007, 20, pp.355-370. �10.1007/s10959- 007-0082-1�. �hal-00015553�

ccsd-00015553, version 1 - 9 Dec 2005

EXPLORATION PROCESS

ROMAIN ABRAHAM AND JEAN-FRANC¸ OIS DELMAS

Abstract. We consider the exploration process associated to the continuous random tree (CRT) built using a L´evy process with no negative jumps. This process has been studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is a useful tool to study CRT as well as super-Brownian motion with general branching mechanism. In this paper we prove this process is Feller, and we compute its infinitesimal generator on exponential functionals and give the corresponding martingale.

1. Introduction

The coding of a L´evy continuous random tree (CRT) by its exploration process can be found in Le Gall and Le Jan [7]. They associate to a L´evy process with no negative jumps that does not drift to +∞, X = (X_t, t ≥0), a critical or sub-critical continuous state branching process (CSBP) and a L´evy CRT which keeps track of the genealogy of the CSBP. The exploration process (ρt, t ≥ 0) takes values in the set of finite measures onR₊. Informally, for an individual t ≥ 0, its generation is recorded by its height H_t = sup Supp ρ_t, where Supp µ stand for the closed support of the measure µ on R. And ρt(dv) records, for the individual labeled t, the “number” of brothers of its ancestor at height v whose labels are larger than t.

For instance, the height process (H_t, t ≥0) of Aldous’ CRT [2] is a normalized Brownian excursion, and the exploration process (ρt, t≥ 0) is the Lebesgue measure, that isρt is the Lebesgue measure on [0, Ht].

The height process, which is enough to code a CRT, is not Markov whereas the exploration process is Markov. The exploration process has been first introduced in order to study super- Brownian motion with general branching mechanism (see [4] and [6]). The strong Markov property of the exploration process is a fundamental property and has been proved in [7]. In [1], the exploration process is used in order to construct a fragmentation process associated to a L´evy process using Markov properties and martingale problems.

The goal of this paper is to give some martingales related to the exploration process. We also prove that the exploration process satisfies the Feller property (and hence has c`ad-l`ag paths and satisfies the strong Markov property). And we compute the generator of the exploration process for exponential functionals. At this point, let us mention that we will always work with the topology of weak convergence for finite measures. But, as the set of finite measures on R₊ is not locally compact for this topology, the Feller property does not make sense on that space. So, we must first extend the definition of the transition function

Date: December 9, 2005.

2000Mathematics Subject Classification. 60J35, 60J80, 60G57.

Key words and phrases. Exploration process, L´evy snake, Feller property, measure valued process, infini- tesimal generator.

The research of the second author was partially supported by NSERC Discovery Grants of the Probability group at Univ. of British Columbia.

1

of the exploration process to the set of finite measures onE =R₊∪ {+∞} endowed with a metric that makes it compact.

We give the general organization of the paper. In Section 2 we recall the construction and definition of the exploration process and some of its properties (which can be found in [4]) that will be used in the sequel. We then state the Feller property, Theorem 2.1. We also give its infinitesimal generator for exponential functionals, Corollary 2.3, as well as the related martingales in Corollaries 2.5 and 2.6. We also recall in Remark 2.8 the relation given in [1]

between the infinitesimal generator of the exploration process associated to the L´evy process with Laplace exponentψand the infinitesimal generator of the exploration process associated to the L´evy process with Laplace exponentψ^(θ)=ψ(θ+·)−ψ(θ),θ≥0. Section 3 is devoted to the proof of the Feller property. We eventually compute the infinitesimal generator of the exploration process for exponential functionals in Section 4.

2. Definitions and main results

2.1. Notations and exploration process. LetS be a metric space, andB(S) its Borelσ- field. We denote byB₊(S) (resp. Bb(S)) the set of real-valued non-negative (resp. bounded) measurable functions defined on S. Let Mf(S) be the set of finite measures defined on S endowed with the topology of weak convergence. We writeMf forMf(R₊). Forµ∈ Mf(S), f ∈ B+(S) we write hµ, fi forR

Sf(x)µ(dx).

We consider aR-valued L´evy processX= (Xt, t≥0) with no negative jumps, no Brownian part and starting from 0. Its law is characterized by its Laplace transform: forλ≥0

E h

e^−λXti

= e^tψ(λ), and we suppose that its Laplace exponent,ψ, is given by

ψ(λ) =α₀λ+ Z

(0,+∞)

π(dℓ)h

e^−λℓ−1 +λℓi ,

with α₀ ≥ 0 and the L´evy measure π is a positive σ-finite measure on (0,+∞) such that Z

(0,+∞)

(ℓ∧ℓ²)π(dℓ) < ∞ and Z

(0,1)

ℓπ(dℓ) = ∞. The first assumption (with the condition α₀ ≥ 0) implies the process X does not drift to +∞, while the second implies X is a.s. of infinite variation.

Forµ∈ Mf, we define

(1) H^µ= sup Supp (µ)∈[0,∞],

where Supp (µ) is the closed support of µ, with the convention sup∅= 0.

We recall the definition and properties of the exploration process which are given in [7], [6] and [4]. The results of this section are mainly extract from [4].

Let I = (I_t, t ≥0) be the infimum process ofX: I_t = inf_0≤s≤tX_s. We will also consider for every 0≤s≤tthe infimum of X over [s, t]:

I_t^s = inf

s≤r≤tX_r.

There exists a sequence (εn, n∈N^∗) of positive real numbers decreasing to 0 s.t.

H˜_t= lim

k→∞

1 εk

Z t

0

1_{Xs<I^s

t+εk} ds exists and is finite a.s. for allt≥0.

The point 0 is regular for the Markov processX−I, and −I is the local time of X−I at 0 (see [3], chap. VII). LetNbe the associated excursion measure of the processX−I out of 0, and σ = inf{t >0;X_t−I_t= 0} be the length of the excursion of X−I underN. Recall that underN,X₀ =I₀ = 0.

From Section 1.2 in [4], there exists a Mf-valued process, ρ⁰ = (ρ⁰_t, t ≥ 0), called the exploration process, such that :

• For each t≥0, a.s. H_t⁰ = ˜H_t, where H_s⁰ =H^ρ0s.

• For every f ∈ B₊(R₊),

hρ⁰_t, fi= Z

[0,t]

d_sI_t^sf(H_s⁰), or equivalently

(2) ρ⁰_t(dr) = X

0<s≤t

X_s−<I_t^s

(I_t^s−Xs−)δ_Hs⁰(dr).

• Almost surely, for every t >0, we have hρ⁰_t,1i=X_t−I_t.

2.2. The Feller property. In Proposition 1.2.3 [4], Duquesne and Le Gall proved that the exploration process is a strong Markov process with c`ad-l`ag paths inMf equipped with the topology of weak convergence (in fact they prove the exploration process is a.s. c`ad-l`ag with the variation distance on finite measures). Here, we improve this result by proving that the exploration process fulfills the Feller property.

We setE =R₊∪ {+∞}equipped with a distance that makesE compact. The setMf(E) is locally compact.

In the definition of the exploration process, as X starts from 0, we have ρ₀ = 0 a.s. To get the Markov property of ρ, we must define the process ρ started at any initial measure µ∈ Mf(E).

Letµ∈ Mf(E). For a∈[0,hµ,1i], we define the erased measure kaµ by kaµ([0, r]) =µ([0, r])∧(hµ,1i −a), forr∈E.

Ifa >hµ,1i, we set kaµ= 0. In other words, the measure kaµis the measure µ erased by a massabackward fromH^µ, with the natural extension of (1) to Mf(E).

Forν, µ∈ Mf(E), we define the concatenation [µ, ν]∈ Mf(E) of the two measures, using the convention x+∞= +∞ forx∈E, by

[µ, ν], f

= µ, f

+

ν, f(H^µ+·)

, f ∈ B+(E).

Eventually, we set for every µ∈ Mf(E) and every t >0, ρt=

k_−Itµ, ρ⁰_t].

We say thatρ= (ρt, t≥0) is the exploration process started atρ₀=µ, and write P_µfor its law. We set H_t = H^ρt. The process ρ is an homogeneous Markov process (this is a direct consequence of Proposition 1.2.3 in [4]).

Theorem 2.1. The exploration process, ρ, defined on Mf(E) equipped with the topology of weak convergence is Feller.

The proof of this theorem is given in Section 3. Sinceρ has the Feller property we recover it enjoys the strong Markov property and has a.s. c`ad-l`ag paths. Notice that ifµ∈ Mf, asρ⁰ isMf valued, we get by construction thatρ isMf valued. We recover that the exploration

process inMf enjoys the strong Markov property and has a.s. c`ad-l`ag paths. We can consider the infinitesimal generator ofρ.

2.3. The infinitesimal generator. Letf a bounded non-negative function defined onR₊ of class C¹ with bounded first derivative such thatf(∞) = lim_t→∞f(t) exists.

We considerF, K ∈ B(Mf(E)) defined by: for µ∈ Mf(E), (3) F(µ) = e^−hµ,fi, and K(µ) =F(µ)

ψ(f(H^µ))−f^′(H^µ)1_{H^µ_<∞}

. Notice that F is a continuous function, whereas K belongs only to B(Mf(E)) a priori.

We denote by Pt the transition function of the exploration process ρ at time t and we consider its resolvent: for λ >0,U_λ =R∞

0 e^−λtP_t. Theorem 2.2. Let λ >0. We have

U_λ(λF −K)(µ) = e^−hµ,fi− f(0)

ψ⁻¹(λ)e^{−ψ−1(λ)hµ,1i}.

The proof of this Theorem is given in Section 4. There exists a local time at µ = 0 which has to be considered when computing the infinitesimal generator. This translates into the condition f(0) = 0 for the domain of the infinitesimal generator, as stated in the next Corollary.

Corollary 2.3. Assume that f(0) = 0, λ > 0. We have U_λ(λF −K) =F. In particular (F, K) belongs to the extended domain of the infinitesimal generator of ρ.

Remark 2.4. Notice the function K is not continuous. However, it can be proved directly that ifµ6= 0 with compact support then

λ→∞lim λ

λU_λF−F

(µ) =K(µ) and if µ= 0

λ→∞lim λ

λUλF −F+ f(0) ψ⁻¹(λ)

(0) =ψ(f(0))−f^′(0).

Standard results on Markov process implies the following Corollary, see e.g. [5] Chapter 4.

Corollary 2.5. Assume f(0) = 0, λ≥0. The process (Mt, t≥0) defined by Mt= e^{−λt−hρt,fi}+

Z t

0

e^{−λs−hρs,fi}

λ−ψ(f(Hs)) +f^′(Hs)1_{Hs<∞}

ds, where Hs=H^ρs, is a martingale w.r.t. the filtration generated by ρ.

We deduce from the optional stopping Theorem and the monotone class Theorem the next result. (Notice we don’t assumef(0) = 0.)

Corollary 2.6. The process (Mt, t≥0) defined by M_t= e^{−hρt∧σ,fi}−

Z t∧σ 0

e^−hρs,fi

ψ(f(H_s))−f^′(H_s)1_{Hs<∞}

ds

is a martingale w.r.t. the filtration generated byρ.

Remark 2.7. Notice from Definition 1.2.1 in [4] thatH_t<∞ P_µ-a.s. for all t >0 ifµ∈ Mf. Hence the indicator1_{Hs<∞}can be removed in Corollaries 2.5 and 2.6 underP_µforµ∈ Mf.

Remark 2.8. From [1] (see Theorem 6.1 and Proposition 6.3 in Section 6.1), there is a re- lationship between the infinitesimal generator of ρ and the infinitesimal generator of the exploration process,ρ^(θ), associated to a L´evy process with exponentψ^(θ), where

ψ^(θ)(λ) =ψ(λ+θ)−ψ(θ), λ≥0.

More precisely, let F, K ∈ B(Mf) bounded such that we have E_µ Z σ

0

|K(ρs)| ds

<∞ for anyµ∈ Mf andMt=F(ρ_t∧σ)−R_t∧σ

0 K(ρs), fort≥0, defines a martingale underP_µ. Then, we have for allµ∈ Mf,P_µ-a.s.

(4)

Z σ^(θ)

0

du Z

(0,∞)

1−e^−θℓ

π(dℓ)

F([ρ^(θ)_u , ℓδ₀])−F(ρ^(θ)_u ) <∞,

whereσ^(θ) = inf{t >0;ρ^(θ)_t = 0}, and the process (Nt, t≥0) is a martingale, where fort≥0, N_t=F(ρ^(θ)

t∧σ^(θ))−

Z t∧σ^(θ) 0

du K(ρ^(θ)_u ) + Z

(0,∞)

1−e^−θℓ

π(dℓ)

F([ρ^(θ)_u , ℓδ₀])−F(ρ^(θ)_u )

! . Now, if we takeF and K defined by (3), and assume thatf ≥εfor some constant ε >0.

Then, we have E_µ

Z σ

0

|K(ρs)| ds

≤E_µ Z σ

0

e^−εhρs,1i ds

= 1−e^−εhµ,1i ψ(ε) <∞.

Notice that F([µ, ℓδ₀])−F(µ)≤0 and Z

(0,∞)

1−e^−θℓ

π(dℓ)

F([µ, ℓδ₀])−F(µ)

=−F(µ) Z

(0,∞)

1−e^−θℓ 1−e^−ℓf(Hµ) π(dℓ)

=−F(µ)

ψ(f(H^µ))−ψ^(θ)(f(H^µ)) . Condition (4) is also satisfied. We get for the martingale (Nt, t≥0) given above that N_t=F(ρ^(θ)

t∧σ^(θ))−

Z t∧σ^(θ) 0

du K(ρ^(θ)_u ) + Z

(0,∞)

1−e^−θℓ

π(dℓ)

F([ρ^(θ)_u , ℓδ₀])−F(ρ^(θ)_u )

!

=F(ρ^(θ)

t∧σ^(θ))−

Z t∧σ^(θ) 0

du F(ρ_s)h

ψ^(θ)(f(H^ρ(θ)s ))−f^′(H^ρ(θ)s )i .

And thanks to Remark 2.7, we recover Corollary 2.6 for the processρ^(θ)for any initial measure inMf.

3. Proof of Theorem 2.1 In this sequelx₊= max(x,0) denotes the positive part ofx∈R.

3.1. A distance which induces the topology of weak convergence. Let G be an increasing non-negative bounded function of class C¹ on R₊. For convenience, we write G(∞) for the limit of G(t) as t goes to infinity, and we shall assume that G(0) = 0 and G(∞) ≤ 1. The set E = R₊∪ {∞} endowed with the distance d(x, y) = |G(x)−G(y)| is compact. Then the set of finite measures onE,Mf(E), equipped with the topology of weak convergence is locally compact.

Forr∈R₊,µ∈ Mf(E), we setHr^µ=H^krµ. Notice the functionr7→Hr^µis non-increasing, right continuous, satisfiesH_hµ,1i^µ = 0. Forr∈[0,hµ,1i], we haveHr^µ=g (hµ,1i−r)⁻

, where

g is the right continuous inverse of the cumulative distribution function of µ. In particular, if forµ, ν ∈ Mf(E) we have hµ,1i=hν,1i and Hr^µ=H_r^ν for all r≥0, then µ=ν.

Lemma 3.1. For all r, v ∈R₊, µ∈ Mf(E), we have (5) v < H_r^µ⇐⇒µ((v,+∞])> r.

For any h∈ Bb(E), we have (6)

Z _hµ,1i

0

h(H_r^µ)dr=hµ, hi.

and if h(0) = 0 this can be also written as (7)

Z ∞ 0

h(H_r^µ)dr=hµ, hi.

Proof. Notice that Hr^µ = sup{x ∈ E;µ([x,+∞]) > r}. This implies (5). Let h be a non- negative non-decreasing bounded function defined onR₊ of class C¹ such thath(0) = 0 and Z

R+

h^′(v)

µ([v,+∞])dv is finite. We have Z _∞

0

h(H_r^µ)dr= Z

R²₊

h^′(v)1_{v<Hr^µ_}drdv

= Z

R²₊

h^′(v)1{r<µ((v,+∞])} drdv

= Z ∞

0

h^′(v)µ((v,+∞])dv

= Z

R+×E

h^′(v)1_{u>v} dvµ(du)

=hµ, hi,

where we used (5) for the second equality, and h(0) = 0 for the first and last equality. By linearity and monotone class Theorem, we get (7) for any h ∈ Bb(E), such that h(0) = 0.

Then (6) is a direct consequence of (7) asHr^µ= 0 forr >hµ,1i.

In particular, (7) holds for h = G. Thus we have R∞

0 G(H_r^µ)dr ≤ hµ,1i < ∞ for all µ∈ Mf(E). We introduce the following function defined onMf(E)²: for µ, ν ∈ Mf(E)

D(µ, ν) =|hµ,1i − hν,1i|+ Z _∞

0

d(H_r^µ, H_r^ν)dr

=|hµ,1i − hν,1i|+

Z max(hµ,1i,hν,1i) 0

d(H_r^µ, H_r^ν)dr.

Since Gis bounded by 1, we have

(8) hµ,1i ≤D(0, µ)≤2hµ,1i.

From what precedes Lemma 3.1, we get that D(µ, ν) = 0 implies µ =ν. It then is easy to check thatD is a distance onMf(E).

Remark 3.2. We deduce from (8) that a sequence (µn, n≥1) is unbounded in (Mf(E), D) if and only if the sequence (hµn,1i, n≥1) is unbounded.

Lemma 3.3. The topology induced by D on Mf(E) corresponds to the topology induced by the weak convergence.

Notice that (Mf(E), D) is a locally compact Polish space.

Proof. We first consider a sequence (µn, n ≥ 1) of elements of Mf(E) which converges for the distanceDtoµ. We shall prove that this sequence converges weakly toµ. Letf ∈ Bb(E) such that|f(x)−f(y)| ≤d(x, y) for all x, y∈E. We set f^∗ forf−f(0). Thanks to (7), we have

|hµn, fi − hµ, fi|=

hµn, f(0)i − hµ, f(0)i+ Z ∞

0

[f^∗(H_r^µn)−f^∗(H_r^µ)]dr

≤ |f(0)||hµn,1i − hµ,1i|+ Z ∞

0

|f^∗(H_r^µn)−f^∗(H_r^µ)|dr

≤ |f(0)||hµn,1i − hµ,1i|+ Z ∞

0

d(H_r^µn, H_r^µ)dr

≤(|f(0)|+ 1)D(µn, µ).

As this holds for any Lipschitz function, we get that (µn, n≥1) converges weakly toµ.

Let (µn, n ≥ 1) be a sequence of elements of Mf(E) which converges weakly to µ. In particular, the sequence (hµn,1i, n ≥ 1) converges to hµ,1i. Thus, there exists a finite constant A such thathµ,1i ≤A andhµn,1i ≤Afor all n≥1. Then, we have

Z ∞ 0

d(H_r^µn, H_r^µ)dr= Z ∞

0

dr|G(H_r^µn)−G(H_r^µ)|

= Z ∞

0

dr

Z ∞ 0

dv G^′(v)[1_{{r<µn((v,∞])}}−1{r<µ((v,∞])}]

≤ Z

R²₊

drdv G^′(v)|1_{{r<µn((v,∞])}}−1{r<µ((v,∞])}|

= Z

R+

dv G^′(v)|µn((v,∞])−µ((v,∞])|,

where we used (5) for the second equality. The weak convergence of (µ_n, n≥ 1) towards µ implies that dv-a.e., lim_n→∞µn((v,∞]) = µ((v,∞]). As G^′(v)|µn((v,∞])−µ((v,∞])| ≤ 2AG^′(v) and 2AG^′ is integrable, we deduce from dominated convergence Theorem that

n→∞lim Z ∞

0

d(H_r^µn, H_r^µ) dr = 0. This and the convergence of (hµn,1i, n ≥ 1) to hµ,1i imply that lim

n→∞D(µn, µ) = 0.

We introduce the distance Din order to get good continuity properties for the concatena- tion and the erasing functions ka.

Lemma 3.4. Let a≥0. We have D(k_aµ, k_aν)≤D(µ, ν).

Proof. Notice Hr^kaρ=H_r+a^ρ . We have

D(k_aµ, k_aν) =|(hµ,1i −a)₊−(hν,1i −a)₊|+ Z _∞

0

d(H_r+a^µ , H_r+a^ν )dr

=|(hµ,1i −a)₊−(hν,1i −a)₊|+ Z _∞

a

d(H_r^µ, H_r^ν)dr

≤D(µ, ν),

where we used for the last inequality the fact that the function x 7→ (x−a)+ is Lipschitz

with constant equal to one.

With the convention x+∞=∞for all x∈E, we have H_r^[kaµ,ρ]=

(Ha^µ+Hr^ρ ifr≤ hρ,1i, H_a+r−hρ,1i^µ ifr≥ hρ,1i.

Let µ ∈ Mf(E), and Aµ ⊂ R₊ be the set of continuity points of the function r 7→ Hr^µ. Notice that R₊\Aµ is at most countable.

Lemma 3.5. Let(µn, n≥1)be a sequence ofMf(E)which converges weakly toµ∈ Mf(E).

For all a∈ Aµ, ρ∈ Mf(E), we have that ([k_aµ_n, ρ], n≥1) converges weakly to [k_aµ, ρ]:

n→∞lim D([kaµn, ρ],[kaµ, ρ]) = 0.

Proof. Let (µ_n, n≥1) be a sequence of elements ofMf(E) which converges to µ∈ Mf(E) for the distance D (i.e. which converges weakly). For all a ≥ 0, we have h[kaµn, ρ],1i = hρ,1i+ (hµn,1i −a)₊ and lim_n→∞h[kaµn, ρ],1i=h[kaµ, ρ],1i. There exists a finite constant A such thathµ,1i ≤Aand hµn,1i ≤A for all n≥1. Furthermore the sequence of functions (G(H_·^µn), n ≥ 1) converges in L¹([0, A], dr) to G(H_·^µ). Since G(H_·^µn) and G(H_·^µ) are non- increasing, we deduce that lim_n→∞G(Ha^µn) = G(Ha^µ) for any continuity point a of G(H_·^µ).

Since G is increasing and continuous, we deduce that for any a ∈ Aµ ∩[0, A], we have lim_n→∞Ha^µn =Ha^µ. The results is also trivially true for a > A.

Leta∈ Aµ. Forr ≤ hρ,1i, we have

H_r^[kaµn,ρ]=H_a^µn+H_r^ρ−−−→

n→∞ H_a^µ+H_r^ρ=H_r^[kaµ,ρ], and for r≥ hρ,1i,dr-a.e.

H_r^[kaµn,ρ]=H_a+r−hρ,1i^µn −−−→

n→∞ H_a+r−hρ,1i^µ =H_r^[kaµ,ρ].

Notice that for r ≥A+hρ,1i, we have Hr^[kaµn,ρ] =Hr^[kaµ,ρ] = 0. Since G is continuous, we get by dominated convergence fora∈ Aµ, that

n→∞lim Z ∞

0

d(H_r^[kaµn,ρ], H_r^[kaµ,ρ])dr= 0.

This implies that fora∈ Aµ, lim_n→∞D([kaµn, ρ],[kaµ, ρ]) = 0.

Let C0(Mf(E)) be the set of real continuous functions defined on (Mf(E), D) such that

n→∞lim F(µn) = 0 whenever lim

n→∞D(0, µn) = +∞, that is whenever lim

n→∞hµn,1i=∞.

Corollary 3.6. Let (I^∗, ρ) be a random variable with values in R₊,×Mf(E), such that the distribution of I^∗ has no atom (i.e. P(I^∗ =x) = 0 for all x ∈ R₊). Let F ∈ C0(Mf(E)).

Then the function F :µ7→ E[F([k_I^∗µ, ρ])] belongs toC0(Mf(E)).

Proof. LetQbe the distribution ofI^∗ and (P_a(dρ), a≥0) be a measurable version of the con- ditional law ofρknowingI^∗: Pa(dρ)Q(da) is the distribution of (I^∗, ρ). ForF ∈ C₀(Mf(E)), we have

E[F([kI^∗µ, ρ])] = Z

Pa[F([kaµ, ρ])]Q(da).

Let (µ_n, n ≤ 1) be a sequence converging to µ. From Lemma 3.5, for all a ∈ Aµ,

n→∞lim D([kaµn, ρ],[kaµ, ρ]) = 0. Since the complementary of Aµ is at most countable and

sinceQ(da) has no atoms, we get thatQ(da)-a.s. lim

n→∞F([kaµn, ρ]) =F([kaµ, ρ]), for anyρ∈ Mf(E). NoticeF is bounded. By dominated convergence, we get that lim

n→∞E[F([kI^∗µn, ρ])] = E[F([kI^∗µ, ρ])]. Thus the functionF is continuous.

Notice that the total mass of [k_I^∗µ, ρ] is equal to (hµ,1i −I^∗)₊+hρ,1i. If lim

n→∞hµn,1i=∞, then a.s. we have limn→∞F([kI^∗µn, ρ]) = 0. By the dominated convergence Theorem, we get lim_n→∞F(µn) = 0.

In conclusion, we get that F ∈C0(Mf(E)).

3.2. Feller property: Proof of Theorem 2.1. Let (P_t, t≥0) be the transition semi-group of the Markov processρ on Mf(E).

RecallC0(Mf(E)) is the set of real continuous functions defined on (Mf(E), D) such that

n→∞lim F(µn) = 0 whenever lim

n→∞D(0, µn) = +∞, that is whenever lim

n→∞hµn,1i=∞.

Let us recall that (see e.g. [8], Proposition III.2.4)ρ is a Feller process if and only if (i) IfF ∈ C0(Mf(E)), then for everyt≥0,PtF ∈ C0(Mf(E)).

(ii) For every F ∈ C0(Mf(E)), for everyµ∈ Mf(E), lim

t→0PtF(µ) =F(µ).

To prove Condition (i), we state the next Lemma.

Lemma 3.7. For every t >0, the distribution of I_t has no atom.

Proof. Let us denote byτ = (τr, r≥0) the right-continuous inverse of the process (−It, t≥0).

We now (cf [3], Chap. VII) that the processτ is a subordinator with Laplace exponentψ⁻¹. As lim_λ→∞ψ(λ)/λ=∞, we have that limλ→∞ψ⁻¹(λ)/λ= 0. The process τ has no drift. It is moreover strictly increasing since the process −It is continuous. It is easy to check that if P(−It =x)> 0 for some x >0, then we have P(−It =x, τ_x =t) >0 and P(τ_−It =t) >0.

This is in contradiction to the fact that τ has no drift (see Theorem 4 in [3]). Hence the

Lemma is proved.

LetF ∈ C0(Mf(E)) and lett≥0. Then, as the distribution of the random variableIthas no atoms (see Lemma 3.7), we can apply Corollary 3.6 with I^∗ =−It and get the function F :µ7→E_µ[F(ρ^µ_t)] belongs toC₀(Mf(E)).

It remains to prove Condition (ii). Let us remark that the exploration process is right- continuous at t= 0. Indeed, for every continuous function f on E bounded by 1, we have, P_µ-a.s.

|hρt, fi − hµ, fi|=

k−Itµ, ρ⁰_t , f

− hµ, fi

=

− hµ−k_−Itµ, fi+

ρ⁰_t, f(H_−I^µ _t+·)

≤ hµ−k_−Itµ,1i+hρ⁰_t,1i

≤ −It+Xt−It.

And the right continuity of ρ at 0 follows from the right-continuity of X and I at 0. Let F ∈ C0(Mf(E)). As the functionF is bounded, we get by dominated convergence that, for everyµ∈ Mf(E),

E_µ[F(ρt)]−−→

t→0 F(µ).

4. Proof of Theorem 2.2

4.1. Other properties of the exploration process. In this section we recall some prop- erties of the exploration process.

4.1.1. Poisson decomposition. Let µ ∈ Mf(E). We decompose the path of ρ under P_µ according to excursions of the total mass ofρ above its minimum. More precisely, we denote by (α_i, β_i), i∈ I the excursion intervals of X−I away from 0. For every i∈ I,t∈(α_i, β_i), we have ραi =k_−Iαiµ=k_−Iαiρt. We defineρⁱ by the formulaρⁱ_t =ρ⁰_(α

i+t)∧βi or equivalently ifµ∈ Mf:

[k_−Iαiµ, ρⁱ_t] =ρ_{(αi+t)∧βi}.

The local time ofX−I at level 0 is given by −I, and if (τr, r ≥0) is the right continuous inverse ofI, then the process (τ_r, r≥0) is a subordinator with Laplace exponentψ⁻¹ (cf [3], Chap. VII). Excursion theory for X−I ensures the following result.

Lemma 4.1. Let µ∈ Mf(E). The point measure X

i∈I

δ_(−Iαi,ρⁱ₎ is under P_µ a Poisson point measure with intensity drN[dρ].

LetF be a non-negative measurable function defined onR₊× Mf(E)×D([0,∞),Mf(E)), where D([0,∞),Mf(E)) stands for the Skorohod space of c`ad-l`ag path in Mf(E). We have

(9) E_µ

"

X

i∈I

F(α_i, ρ_αi, ρⁱ)

#

=E_µ Z ∞

0

drN[F(v, k_rµ, ρ)]_|v=τ

r

.

4.1.2. The dual process and representation formula. We shall need the Mf-valued process η= (ηt, t≥0) defined underNby

ηt(dr) = X

0<s≤t

Xs−<I_t^s

(Xs−I_t^s)δHs(dr).

The processη is the dual process of ρ under N(see Corollary 3.1.6 in [4]). Let σ = inf{s >

0;ρs = 0} denote the length of the excursion. Using this duality, it easy to check (see the proof of Lemma 3.2.2 in [4]), that for any bounded measurable functionF defined onMf,

(10) N

Z σ

0

e^−ψ(γ)tF(ρt)

=N Z σ

0

e^−γhηt,1iF(ρt)

.

We recall the Poisson representation of (ρ, η) underN. LetN(dx dℓ du) be a Poisson point measure on [0,+∞)³ with intensity

dx ℓπ(dℓ)1_[0,1](u)du.

For every a > 0, let us denote by M_a the law of the pair (µa, νa) of finite measures on R₊ defined by: forf ∈ B+(R₊)

hµa, fi= Z

N(dx dℓ du)1_[0,a](x)uℓf(x), (11)

hνa, fi= Z

N(dx dℓ du)1_[0,a](x)ℓ(1−u)f(x).

(12)

We eventually set M=R_+∞

0 da e^−α0aM_a.

Proposition 4.2. For every non-negative measurable function F on M²_f,

N Z σ

0

F(ρt, ηt)dt

= Z

M(dµ dν)F(µ, ν).

4.2. Computation of the resolvent. Let f be a bounded function defined on R₊ of class C¹ with bounded first derivative such that f(∞) = limt→∞f(t) exists. By convention, we putf^′(∞) = 0.

We set for µ∈ Mf(E)

(13) Fλ(µ) = e^−hµ,fi λ−ψ(f(H^µ)) +f^′(H^µ) .

Letγ =ψ⁻¹(λ). We shall compute U_λ(F_λ)(µ) =E_µ

Z ∞ 0

dt e^{−ψ(γ)t−hρt,fi} ψ(γ)−ψ(f(Ht)) +f^′(Ht)

.

We define the function f_r by f_r(·) = f(·+Hr^µ), where Hr^µ = H^krµ and use the convention x+∞=x for all x∈E. Using notations of Section 4.1.1 and (9), we have

Uλ(Fλ)(µ) =E_µ

"

X

i∈I

e^−ψ(γ)αi

Z βi−αi

0

dt e^{−ψ(γ)t−h[k−Iαiµ,ρit],fi} ψ(γ)−ψ(f_−Iαi(H^ρit)) +f_−I^′

αi(H^ρit)i

=E_µ Z ∞

0

dr e^−ψ(γ)τrN Z σ

0

dt e^{−ψ(γ)t−h[krµ,ρt],fi} ψ(γ)−ψ(fr(Ht)) +f_r^′(Ht)

. Recall (τr, r≥0) is a subordinator with Laplace exponentψ⁻¹and (10). Sinceh[krµ, ρt], fi= hkrµ, fi+hρt, f_ri, we get

Uλ(Fλ)(µ) = Z ∞

0

dr e^−γre^−hkrµ,fiN Z σ

0

dt e^{−γhηt,1i−hρt,fri} ψ(γ)−ψ(fr(Ht)) +f_r^′(Ht)

.

We deduce from Proposition 4.2 that N

Z σ

0

dt e^{−γhηt,1i−hρt,fri} ψ(γ)−ψ(f_r(H_t)) +f_r^′(H_t)

= Z _∞

0

da e^−α0a ψ(γ)−ψ(fr(a)) +f_r^′(a) exp

(

− Z a

0

dx Z 1

0

du Z

(0,∞)

ℓπ(dℓ)h

1−e−γ(1−u)ℓ−ufr(x)ℓi )

= Z _∞

0

da ψ(γ)−ψ(f(a+H_r^µ)) +f^′(a+H_r^µ)

e^{−R0adxR01du ψ′(γ(1−u)+uf(x+Hrµ))}. We set for y∈E

Λ(y) = Z _∞

0

da ψ(γ)−ψ(f(a+y)) +f^′(a+y)

e^−g(a,y), whereg(a, y) =

Z a

0

dx ψ(f(x+y))−ψ(γ)

f(x+y)−γ with the convention ψ(v)−ψ(v)

v−v =ψ^′(v), so that

(14) U_λ(F_λ)(µ) =

Z ∞ 0

dr e^−γre^−hkrµ,fiΛ(H_r^µ).

Sinceψis positive increasing continuous andfbounded, we have lima→∞g(a, y) =∞. Notice that fory ∈[0,∞)

Λ(y) = Z _∞

0

da −∂ag(a, y)(f(a+y)−γ) +f^′(a+y)

e^−g(a,y)

=h

(f(a+y)−γ) e^−g(a,y)i∞ 0

=γ −f(y).

We also have Λ(∞) =γ−f(∞). We deduce from (14) that U_λ(F_λ)(µ) =

Z _∞

0

dr e^{−γr−hkrµ,fi}(γ−f(H_r^µ)).

Notice that hkrµ, fi= 0 and Hr^µ= 0 for r >hµ,1i, so that we have Z _∞

hµ,1i

dr e^{−γr−hkrµ,fi}(γ−f(H_r^µ)) = Z _∞

hµ,1i

dr e^−γr(γ−f(0)) = e^−γhµ,1i

1−f(0) γ

.

We deduce from (6) withµreplaced by k_rµthat for r≤ hµ,1i, hkrµ, fi=

Z _hµ,1i−r

0

f(H_v^krµ)dv = Z _hµ,1i

r

f(H_v^µ)dv.

This implies that Z hµ,1i

0

dr e^{−γr−hkrµ,fi}(γ−f(H_r^µ)) = Z hµ,1i

0

dr e^{−γr−Rrhµ,1if(Hµv)dv}(γ−f(H_r^µ))

=h

−e^{−γr−Rrhµ,1if(Hvµ)dv}ihµ,1i 0

= e^−hµ,fi−e^−γhµ,1i,

where we used (6) for the first term of the right hand side of the last equation. Eventually, we have

Uλ(Fλ)(µ) = e^−hµ,fi−f(0)

γ e^−γhµ,1i. References

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[2] D. ALDOUS. The continuum random tree III.Ann. Probab., 21(1):248–289, 1993.

[3] J. BERTOIN.L´evy processes. Cambridge University Press, Cambridge, 1996.

[4] T. DUQUESNE and J.-F. LE GALL.Random trees, L´evy processes and spatial branching processes, volume 281. Ast´erisque, 2002.

[5] S. N. ETHIER and T. G. KURTZ.Markov processes. Wiley, 1986.

[6] J.-F. LE GALL and Y. LE JAN. Branching processes in L´evy processes: Laplace functionals of snake and superprocesses. Ann. Probab., 26:1407–1432, 1998.

[7] J.-F. LE GALL and Y. LE JAN. Branching processes in L´evy processes: The exploration process.Ann.

Probab., 26:213–252, 1998.

[8] D. REVUZ and M. YOR.Continuous martingales and Brownian motion. Springer Verlag, 2nd edition, 1995.

Romain Abraham, MAPMO, Universit´e d’Orl´eans, B.P. 6759, 45067 Orl´eans cedex 2 France E-mail address: romain.abraham@univ-orleans.fr

Jean-Franc¸ois Delmas, ENPC-CERMICS, 6-8 av. Blaise Pascal, Champs-sur-Marne, 77455 Marne La Vall´ee, France.

E-mail address: delmas@cermics.enpc.fr